They need a warrant to physically alter or attach things to private property to track them. They don't need a warrant to post monitors and record what they see in plain view in public spaces with no expectation of privacy. The entire concept of a license plate, universal around the world, works against the idea that the state can't monitor your car.

I'm not saying ALPRs don't pose new privacy problems and, in the long term, depending on how they're used, even constitutional problems. But clearly the Anti-Pinkerton Act doesn't get you anywhere here.

It depends on how. The Fourth Amendment prohibits warrantless searches, not any information itself. The police can always just have an officer tail you 24/7, and it's perfectly legal. Placing a GPS tracker on your car physically invades your property and therefore counts as a search though. Generally any public photography is not a search, so they're free to record and keep records however they legally can.

Though at some point, even SCOTUS just does whatever feels right, regardless of what the law says. In Carpenter, SCOTUS ruled 5-4 that your cellular company voluntarily handing over historical cell data also counts as a government search. An appellate court has held that if photography is extensive enough, it becomes a search. SCOTUS has held before that uncommon photographic equipment can constitute a search. That logic honestly doesn't really make sense, but it is what it is now. I wouldn't be surprised that the courts rule against it, but that's not what the law really says.